3.73.38 \(\int \frac {-x^2+2 x \log (25)-\log ^2(25)+(15 x^2-20 x \log (25)+5 \log ^2(25)) \log (x)+(-3 x^2+4 x \log (25)-\log ^2(25)) \log (x) \log (\log (x))}{(-2-5 x^3+10 x^2 \log (25)-5 x \log ^2(25)) \log (x)+(x^3-2 x^2 \log (25)+x \log ^2(25)) \log (x) \log (\log (x))} \, dx\)

Optimal. Leaf size=22 \[ -\log \left (2+x (x-\log (25))^2 (5-\log (\log (x)))\right ) \]

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Rubi [F]  time = 21.84, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {-x^2+2 x \log (25)-\log ^2(25)+\left (15 x^2-20 x \log (25)+5 \log ^2(25)\right ) \log (x)+\left (-3 x^2+4 x \log (25)-\log ^2(25)\right ) \log (x) \log (\log (x))}{\left (-2-5 x^3+10 x^2 \log (25)-5 x \log ^2(25)\right ) \log (x)+\left (x^3-2 x^2 \log (25)+x \log ^2(25)\right ) \log (x) \log (\log (x))} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-x^2 + 2*x*Log[25] - Log[25]^2 + (15*x^2 - 20*x*Log[25] + 5*Log[25]^2)*Log[x] + (-3*x^2 + 4*x*Log[25] - L
og[25]^2)*Log[x]*Log[Log[x]])/((-2 - 5*x^3 + 10*x^2*Log[25] - 5*x*Log[25]^2)*Log[x] + (x^3 - 2*x^2*Log[25] + x
*Log[25]^2)*Log[x]*Log[Log[x]]),x]

[Out]

-Log[x] - 2*Log[x - Log[25]] - (Log[625]*Defer[Int][1/(x*(-2 - 5*x^3 + 10*x^2*Log[25] - 5*x*Log[25]^2 + x*(x -
Log[25])^2*Log[Log[x]])), x])/Log[25] - Log[25]^2*Defer[Int][1/(Log[x]*(-2 - 5*x^3 + 10*x^2*Log[25] - 5*x*Log
[25]^2 + x*(x - Log[25])^2*Log[Log[x]])), x] + 2*Log[25]*Defer[Int][x/(Log[x]*(-2 - 5*x^3 + 10*x^2*Log[25] - 5
*x*Log[25]^2 + x*(x - Log[25])^2*Log[Log[x]])), x] - Defer[Int][x^2/(Log[x]*(-2 - 5*x^3 + 10*x^2*Log[25] - 5*x
*Log[25]^2 + x*(x - Log[25])^2*Log[Log[x]])), x] - 6*Defer[Int][1/((x - Log[25])*(-2 - 5*x^3 + 10*x^2*Log[25]
- 5*x*Log[25]^2 + x^3*Log[Log[x]] - 2*x^2*Log[25]*Log[Log[x]] + x*Log[25]^2*Log[Log[x]])), x] + (Log[625]*Defe
r[Int][1/((x - Log[25])*(-2 - 5*x^3 + 10*x^2*Log[25] - 5*x*Log[25]^2 + x^3*Log[Log[x]] - 2*x^2*Log[25]*Log[Log
[x]] + x*Log[25]^2*Log[Log[x]])), x])/Log[25]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {(x-\log (25)) (x-\log (25)+(3 x-\log (25)) \log (x) (-5+\log (\log (x))))}{\log (x) \left (2+5 x^3-10 x^2 \log (25)+5 x \log ^2(25)-x (x-\log (25))^2 \log (\log (x))\right )} \, dx\\ &=\int \left (\frac {-3 x+\log (25)}{x (x-\log (25))}+\frac {-x^4+3 x^3 \log (25)-3 x^2 \log ^2(25)+x \log ^3(25)-6 x \log (x)+\log (625) \log (x)}{x (x-\log (25)) \log (x) \left (-2-5 x^3+10 x^2 \log (25)-5 x \log ^2(25)+x^3 \log (\log (x))-2 x^2 \log (25) \log (\log (x))+x \log ^2(25) \log (\log (x))\right )}\right ) \, dx\\ &=\int \frac {-3 x+\log (25)}{x (x-\log (25))} \, dx+\int \frac {-x^4+3 x^3 \log (25)-3 x^2 \log ^2(25)+x \log ^3(25)-6 x \log (x)+\log (625) \log (x)}{x (x-\log (25)) \log (x) \left (-2-5 x^3+10 x^2 \log (25)-5 x \log ^2(25)+x^3 \log (\log (x))-2 x^2 \log (25) \log (\log (x))+x \log ^2(25) \log (\log (x))\right )} \, dx\\ &=\int \left (-\frac {1}{x}-\frac {2}{x-\log (25)}\right ) \, dx+\int \frac {x (x-\log (25))^3-(-6 x+\log (625)) \log (x)}{x (x-\log (25)) \log (x) \left (2+5 x^3-10 x^2 \log (25)+5 x \log ^2(25)-x (x-\log (25))^2 \log (\log (x))\right )} \, dx\\ &=-\log (x)-2 \log (x-\log (25))+\int \left (\frac {x^4-3 x^3 \log (25)+3 x^2 \log ^2(25)-x \log ^3(25)+6 x \log (x)-\log (625) \log (x)}{x \log (25) \log (x) \left (-2-5 x^3+10 x^2 \log (25)-5 x \log ^2(25)+x^3 \log (\log (x))-2 x^2 \log (25) \log (\log (x))+x \log ^2(25) \log (\log (x))\right )}+\frac {-x^4+3 x^3 \log (25)-3 x^2 \log ^2(25)+x \log ^3(25)-6 x \log (x)+\log (625) \log (x)}{(x-\log (25)) \log (25) \log (x) \left (-2-5 x^3+10 x^2 \log (25)-5 x \log ^2(25)+x^3 \log (\log (x))-2 x^2 \log (25) \log (\log (x))+x \log ^2(25) \log (\log (x))\right )}\right ) \, dx\\ &=-\log (x)-2 \log (x-\log (25))+\frac {\int \frac {x^4-3 x^3 \log (25)+3 x^2 \log ^2(25)-x \log ^3(25)+6 x \log (x)-\log (625) \log (x)}{x \log (x) \left (-2-5 x^3+10 x^2 \log (25)-5 x \log ^2(25)+x^3 \log (\log (x))-2 x^2 \log (25) \log (\log (x))+x \log ^2(25) \log (\log (x))\right )} \, dx}{\log (25)}+\frac {\int \frac {-x^4+3 x^3 \log (25)-3 x^2 \log ^2(25)+x \log ^3(25)-6 x \log (x)+\log (625) \log (x)}{(x-\log (25)) \log (x) \left (-2-5 x^3+10 x^2 \log (25)-5 x \log ^2(25)+x^3 \log (\log (x))-2 x^2 \log (25) \log (\log (x))+x \log ^2(25) \log (\log (x))\right )} \, dx}{\log (25)}\\ &=-\log (x)-2 \log (x-\log (25))+\frac {\int \frac {-x (x-\log (25))^3-(6 x-\log (625)) \log (x)}{x \log (x) \left (2+5 x^3-10 x^2 \log (25)+5 x \log ^2(25)-x (x-\log (25))^2 \log (\log (x))\right )} \, dx}{\log (25)}+\frac {\int \frac {x (x-\log (25))^3-(-6 x+\log (625)) \log (x)}{(x-\log (25)) \log (x) \left (2+5 x^3-10 x^2 \log (25)+5 x \log ^2(25)-x (x-\log (25))^2 \log (\log (x))\right )} \, dx}{\log (25)}\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.80, size = 52, normalized size = 2.36 \begin {gather*} -\log \left (2+5 x^3-10 x^2 \log (25)+5 x \log ^2(25)-x^3 \log (\log (x))+2 x^2 \log (25) \log (\log (x))-x \log ^2(25) \log (\log (x))\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-x^2 + 2*x*Log[25] - Log[25]^2 + (15*x^2 - 20*x*Log[25] + 5*Log[25]^2)*Log[x] + (-3*x^2 + 4*x*Log[2
5] - Log[25]^2)*Log[x]*Log[Log[x]])/((-2 - 5*x^3 + 10*x^2*Log[25] - 5*x*Log[25]^2)*Log[x] + (x^3 - 2*x^2*Log[2
5] + x*Log[25]^2)*Log[x]*Log[Log[x]]),x]

[Out]

-Log[2 + 5*x^3 - 10*x^2*Log[25] + 5*x*Log[25]^2 - x^3*Log[Log[x]] + 2*x^2*Log[25]*Log[Log[x]] - x*Log[25]^2*Lo
g[Log[x]]]

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fricas [B]  time = 0.77, size = 83, normalized size = 3.77 \begin {gather*} -2 \, \log \left (x - 2 \, \log \relax (5)\right ) - \log \relax (x) - \log \left (-\frac {5 \, x^{3} - 20 \, x^{2} \log \relax (5) + 20 \, x \log \relax (5)^{2} - {\left (x^{3} - 4 \, x^{2} \log \relax (5) + 4 \, x \log \relax (5)^{2}\right )} \log \left (\log \relax (x)\right ) + 2}{x^{3} - 4 \, x^{2} \log \relax (5) + 4 \, x \log \relax (5)^{2}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4*log(5)^2+8*x*log(5)-3*x^2)*log(x)*log(log(x))+(20*log(5)^2-40*x*log(5)+15*x^2)*log(x)-4*log(5)^
2+4*x*log(5)-x^2)/((4*x*log(5)^2-4*x^2*log(5)+x^3)*log(x)*log(log(x))+(-20*x*log(5)^2+20*x^2*log(5)-5*x^3-2)*l
og(x)),x, algorithm="fricas")

[Out]

-2*log(x - 2*log(5)) - log(x) - log(-(5*x^3 - 20*x^2*log(5) + 20*x*log(5)^2 - (x^3 - 4*x^2*log(5) + 4*x*log(5)
^2)*log(log(x)) + 2)/(x^3 - 4*x^2*log(5) + 4*x*log(5)^2))

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giac [B]  time = 0.20, size = 51, normalized size = 2.32 \begin {gather*} -\log \left (x^{3} \log \left (\log \relax (x)\right ) - 4 \, x^{2} \log \relax (5) \log \left (\log \relax (x)\right ) + 4 \, x \log \relax (5)^{2} \log \left (\log \relax (x)\right ) - 5 \, x^{3} + 20 \, x^{2} \log \relax (5) - 20 \, x \log \relax (5)^{2} - 2\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4*log(5)^2+8*x*log(5)-3*x^2)*log(x)*log(log(x))+(20*log(5)^2-40*x*log(5)+15*x^2)*log(x)-4*log(5)^
2+4*x*log(5)-x^2)/((4*x*log(5)^2-4*x^2*log(5)+x^3)*log(x)*log(log(x))+(-20*x*log(5)^2+20*x^2*log(5)-5*x^3-2)*l
og(x)),x, algorithm="giac")

[Out]

-log(x^3*log(log(x)) - 4*x^2*log(5)*log(log(x)) + 4*x*log(5)^2*log(log(x)) - 5*x^3 + 20*x^2*log(5) - 20*x*log(
5)^2 - 2)

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maple [B]  time = 0.13, size = 65, normalized size = 2.95

method	result	size
risch	\(-2 \ln \left (x -2 \ln \relax (5)\right )-\ln \relax (x )-\ln \left (\ln \left (\ln \relax (x )\right )-\frac {20 x \ln \relax (5)^{2}-20 x^{2} \ln \relax (5)+5 x^{3}+2}{x \left (4 \ln \relax (5)^{2}-4 x \ln \relax (5)+x^{2}\right )}\right )\)	\(65\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-4*ln(5)^2+8*x*ln(5)-3*x^2)*ln(x)*ln(ln(x))+(20*ln(5)^2-40*x*ln(5)+15*x^2)*ln(x)-4*ln(5)^2+4*x*ln(5)-x^2
)/((4*x*ln(5)^2-4*x^2*ln(5)+x^3)*ln(x)*ln(ln(x))+(-20*x*ln(5)^2+20*x^2*ln(5)-5*x^3-2)*ln(x)),x,method=_RETURNV
ERBOSE)

[Out]

-2*ln(x-2*ln(5))-ln(x)-ln(ln(ln(x))-(20*x*ln(5)^2-20*x^2*ln(5)+5*x^3+2)/x/(4*ln(5)^2-4*x*ln(5)+x^2))

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maxima [B]  time = 0.48, size = 83, normalized size = 3.77 \begin {gather*} -2 \, \log \left (x - 2 \, \log \relax (5)\right ) - \log \relax (x) - \log \left (-\frac {5 \, x^{3} - 20 \, x^{2} \log \relax (5) + 20 \, x \log \relax (5)^{2} - {\left (x^{3} - 4 \, x^{2} \log \relax (5) + 4 \, x \log \relax (5)^{2}\right )} \log \left (\log \relax (x)\right ) + 2}{x^{3} - 4 \, x^{2} \log \relax (5) + 4 \, x \log \relax (5)^{2}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4*log(5)^2+8*x*log(5)-3*x^2)*log(x)*log(log(x))+(20*log(5)^2-40*x*log(5)+15*x^2)*log(x)-4*log(5)^
2+4*x*log(5)-x^2)/((4*x*log(5)^2-4*x^2*log(5)+x^3)*log(x)*log(log(x))+(-20*x*log(5)^2+20*x^2*log(5)-5*x^3-2)*l
og(x)),x, algorithm="maxima")

[Out]

-2*log(x - 2*log(5)) - log(x) - log(-(5*x^3 - 20*x^2*log(5) + 20*x*log(5)^2 - (x^3 - 4*x^2*log(5) + 4*x*log(5)
^2)*log(log(x)) + 2)/(x^3 - 4*x^2*log(5) + 4*x*log(5)^2))

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mupad [F]  time = 0.00, size = -1, normalized size = -0.05 \begin {gather*} \int \frac {4\,{\ln \relax (5)}^2-\ln \relax (x)\,\left (15\,x^2-40\,\ln \relax (5)\,x+20\,{\ln \relax (5)}^2\right )-4\,x\,\ln \relax (5)+x^2+\ln \left (\ln \relax (x)\right )\,\ln \relax (x)\,\left (3\,x^2-8\,\ln \relax (5)\,x+4\,{\ln \relax (5)}^2\right )}{\ln \relax (x)\,\left (5\,x^3-20\,\ln \relax (5)\,x^2+20\,{\ln \relax (5)}^2\,x+2\right )-\ln \left (\ln \relax (x)\right )\,\ln \relax (x)\,\left (x^3-4\,\ln \relax (5)\,x^2+4\,{\ln \relax (5)}^2\,x\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((4*log(5)^2 - log(x)*(20*log(5)^2 - 40*x*log(5) + 15*x^2) - 4*x*log(5) + x^2 + log(log(x))*log(x)*(4*log(5
)^2 - 8*x*log(5) + 3*x^2))/(log(x)*(20*x*log(5)^2 - 20*x^2*log(5) + 5*x^3 + 2) - log(log(x))*log(x)*(4*x*log(5
)^2 - 4*x^2*log(5) + x^3)),x)

[Out]

int((4*log(5)^2 - log(x)*(20*log(5)^2 - 40*x*log(5) + 15*x^2) - 4*x*log(5) + x^2 + log(log(x))*log(x)*(4*log(5
)^2 - 8*x*log(5) + 3*x^2))/(log(x)*(20*x*log(5)^2 - 20*x^2*log(5) + 5*x^3 + 2) - log(log(x))*log(x)*(4*x*log(5
)^2 - 4*x^2*log(5) + x^3)), x)

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sympy [B]  time = 1.28, size = 65, normalized size = 2.95 \begin {gather*} - \log {\relax (x )} - 2 \log {\left (x - 2 \log {\relax (5 )} \right )} - \log {\left (\log {\left (\log {\relax (x )} \right )} + \frac {- 5 x^{3} + 20 x^{2} \log {\relax (5 )} - 20 x \log {\relax (5 )}^{2} - 2}{x^{3} - 4 x^{2} \log {\relax (5 )} + 4 x \log {\relax (5 )}^{2}} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4*ln(5)**2+8*x*ln(5)-3*x**2)*ln(x)*ln(ln(x))+(20*ln(5)**2-40*x*ln(5)+15*x**2)*ln(x)-4*ln(5)**2+4*
x*ln(5)-x**2)/((4*x*ln(5)**2-4*x**2*ln(5)+x**3)*ln(x)*ln(ln(x))+(-20*x*ln(5)**2+20*x**2*ln(5)-5*x**3-2)*ln(x))
,x)

[Out]

-log(x) - 2*log(x - 2*log(5)) - log(log(log(x)) + (-5*x**3 + 20*x**2*log(5) - 20*x*log(5)**2 - 2)/(x**3 - 4*x*
*2*log(5) + 4*x*log(5)**2))

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